According to Nebraska law, persons with STDs are not permitted to marry. The actual writing of the law is “no person who is afflicted with a venereal disease shall marry in this state." Since blood tests are not required to obtain a marriage license, this law may be a little difficult to enforce.
It’s unclear why the law was created. This relatively obscure piece of legislation has been on the books in Nebraska since at least 1944 and has not been amended since the 70s.